Journal of Fourier Analysis and Applications

, Volume 3, Issue 5, pp 499–504

Stability theorems for Fourier frames and wavelet Riesz bases

  • Radu Balan
Article

Abstract

In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis, our estimate reduces to Kadec’ optimal 1/4 result. The second application proves that a phenomenon discovered by Daubechies and Tchamitchian [4] for the orthonormal Meyer wavelet basis (stability of the Riesz basis property under small changes of the translation parameter) actually holds for a large class of wavelet Riesz bases.

Math Subject Classifications

Primary 42C15 Secondary 41A30 

Keywords and Phrases

Frames Riesz basis nonharmonic series wavelets 

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References

  1. [1]
    Christensen, O. (1995). A Paley-Wiener Theorem for Frames.Proc. Am. Math. Soc. 123, 2199–2202.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Christensen, O. and Heil, C., Perturbations of Banach Frames and Atomic Decompositions. To appear inMath. Nach. or http@tyche.mat.univie.ac.at.Google Scholar
  3. [3]
    Cvetković, Z. and Vetterli, M. (1995). Error Analysis in Oversampled A/D Conversion and Quantization of Weyl-Heisenberg Frame Expansions. Mem. No. UCB/ERL M95/48, May.Google Scholar
  4. [4]
    Daubechies, I. (1990). The Wavelet Transform, Time-Frequency Localization and Signal Analysis.IEEE Trans. Inform. Theory 36(5), 961–1005.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Duffin, R.J. and Eachus, J.J. (1942). Some Notes on an Expansion Theorem of Paley and Wiener.Bull. Am. Math. Soc. 48, 850–855.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Duffin, R.J. and Schaeffer, A.C. (1952). A Class of Nonharmonic Fourier Series.Trans. AMS 72, 341–366.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Favier, S.J. and Zalik, R.A. (1995)On the stability of frames and Riesz bases, Appl. Comp. Harmon. Annal.,2(2), 160–173.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Kadec, M.I. (1964). The Exact Value of the Paley-Wiener Constant.Sov. Math. Doklady 5(2), 559–561.Google Scholar
  9. [9]
    Kato, T. (1976).Perturbation Theory for Linear Operators. Springer-Verlag.Google Scholar
  10. [10]
    Levinson, N. (1940). Gap and Density Theorems.AMS. Coll Public. 26.Google Scholar
  11. [11]
    Paley, R.E.A.C. and Wiener, N., (1934). Fourier Transforms in the Complex Domain.AMS Colloq. Publ. vol.19, AMS, Providence RI, reprint 1960.MATHGoogle Scholar
  12. [12]
    Young, R.M. (1980).An Introduction to Nonharmonic Fourier Series. Academic Press, New York.MATHGoogle Scholar

Copyright information

© CRC Press LLC 1997

Authors and Affiliations

  • Radu Balan
    • 1
  1. 1.Program in Applied and Computational MathematicsPrinceton UniversityPrinceton

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